# Algebra and Combinatorics Seminar, Fall 2013

### Wednesdays 3:40-4:30, IES 111

Organizers:  Stephen Doty, Tony Giaquinto, Aaron Lauve, Peter Tingley

 September 11 Monica Vazirani (UC Davis) Representations of the affine BMW algebra 18 Canceled 25 Theo Johnson-Freyd (Northwestern) A properadic approach to the deformation quantization of topological field theories October 2 Peter Tingley (Loyola) PBW bases, multisegements and Young tableaux 9 Canceled 16 Cynthia Vinzant (Michigan) Hyperbolic polynomials, interlacers, and sums of squares. *17* BVM hall 11th floor seminar room Apoorva Khare (Stanford) Faces and maximizer subsets of highest weight modules 23 Sarah Mason (Wake Forest) Quasisymmetric Hook Schur Functions 30 Ben Salisbury (Central Michigan) The Gindikin-Karpelevich formula and marginally large Young tableaux November 6 Steven Sam (UC Berkeley) Infinite rank classical groups Saturday Nov 9 *Note unusual day and location* ALGECOM 9 conference 13 Matt Douglass (North Texas) The free Lie algebra and Schur-Weyl duality 20 Bill Chin (DePaul) Special biserial coalgebras and quantum SL(2) 27 No talk: thanksgiving December 4 Emily Peters The D_2n planar algebras, and knot theories

Directions

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)

Parking is available on-campus for $7 in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue. Abstracts Sept. 11: Monica Vazirani (UC Davis): Representations of the affine BMW algebra. The BMW algebra is a deformation of the Brauer algebra, and has the Hecke algebra of type A as a quotient. Its specializations play a role in types B, C, D akin to that of the symmetric group in Schur-Weyl duality. One can enlarge these algebras by a commutative subalgebra X to an affine, or annular, version. Unlike the affine Hecke algebra, the affine BMW algebra is not of finite rank as a right X-module, so induction functors are ill-behaved, and many of the classical Hecke-theoretic constructions of simple modules fail. However, the affine BMW algebra still has a nice class of X-semisimple, or calibrated, representations, that don't necessarily factor through the affine Hecke algebra. I will discuss Walker's TQFT-motivated 2-handle construction of the X-semisimple, or calibrated, representations of the affine BMW algebra. Sept. 25: Theo Johnson-Freyd (Northwestern): A properadic approach to the deformation quantization of topological field theories. I will describe how Koszul duality and the bar construction for properads is related to the path integral quantization of topological field theories. As an application, I will give a class of Poisson structures that admit a canonical wheel-free deformation quantization. Oct 2: Peter Tingley (Loyola): PBW bases, multisegements and Young tableaux I will report on work done with with our students John Claxton and Tynan Greene over the summer. Specifically, I will discuss the relationship between the realizations of crystal for sl(n) representations using young tableaux and the realization of the infinity crystal using multi-segments. I'll then relate this to PBW bases by showing that, for one well-chosen reduced expression for the longest word in the Weyl group, there is a simple relationship between PBW monomials and multi-segments. Perhaps most interestingly, this gives a simple description of the crystal operators on these PBW monomials, and also a description of which of these PBW basis elements correspond to elements of any given finite crystal. Oct.16: Cynthia Vinzant (Michigan): Hyperbolic polynomials, interlacers, and sums of squares. Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovals, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. I'll give an introduction to this topic and discuss the special connection between hyperbolic polynomials and their interlacing polynomials (whose real ovals interlace the those of the original). This will let us related inner oval of a hyperbolic hypersurface to the cone of nonnegative polynomials and, sometimes, to sums of squares. An important example will be the basis generating polynomial of a matroid. This is joint work with Mario Kummer and Daniel Plaumann. Oct.17: Apoorva Khare (Stanford): Faces and maximizer subsets of highest weight modules. Verma modules over a complex semisimple Lie algebra, as well as their simple quotients are important and well-studied objects in representation theory. We present three formulas to compute the set of weights of all such simple highest weight modules (and others) over a complex semisimple Lie algebra \g. These formulas are direct and do not involve cancellations. Our results extend the notion of the Weyl polytope to general highest weight \g-modules$V^\mu$. We also show that for all such simple modules, the convex hull of the weights is a W_J-invariant polyhedron for some parabolic subgroup W_J. We compute its vertices, faces, and symmetries - more generally, we do so for all parabolic Verma modules, and for all modules$V^\mu$with \mu not on a simple root hyperplane. Our techniques also enable us to completely classify inclusions between "weak faces" of arbitrary$V^\mu\$, in the process extending results of Vinberg, Chari, Cellini, and others from finite-dimensional modules to all highest weight modules.

Oct.23: Sarah Mason (Wake Forest): Quasisymmetric Hook Schur Functions.

Hook Schur functions are a class of symmetric functions (in two sets of variables) introduced in 1987 by Berele and Regev to prove a generalization of Weyl's "Strip Theorem". We introduce a generalization of hook Schur functions to the space of quasisymmetric functions and prove that several useful properties of the hook Schur functions remain true in the quasisymmetric setting.

Oct.30: Ben Salisbury (Central Michigan): The Gindikin-Karpelevich formula and marginally large Young tableaux.

The Gindikin-Karpelevich formula determines the normalizing constant of an intertwining map between two induced representations of a split reductive group G. There have been recent developments by Brubaker-Bump-Friedberg, Bump-Nakasuji, and others expressing this constant as a sum over a crystal graph, stemming from work on Weyl group multiple Dirichlet series. In this talk, I will explain some of the recent work relating the Gindikin-Karpelevich formula to crystals solely from the combinatorial perspective and give a description of the formula using the marginally large tableaux defined by J. Hong and H. Lee. The latter is joint work with Kyu-Hwan Lee.

Nov 6: Steven Sam (Berkeley) Infinite rank classical groups.

The category of polynomial representations of the infinite rank (complex) general linear group is the universal setting for doing tensor constructions and is closely related to the combinatorics of symmetric functions. I'll explain some joint work with Andrew Snowden on the categories of representations of the infinite rank orthogonal and symplectic groups. It turns out these are the universal settings for doing tensor constructions with the additional data of a bilinear form and is an algebraic setting for understanding stable combinatorial formulas involving the classical groups (such as branching rules, tensor product multiplicities). I'll also explain some joint work with Andrew Snowden and Jerzy Weyman on the relation between infinite rank and finite rank classical groups which explains the unstable behavior of these formulas which was previously studied by Koike and Terada.

Nov 13: Matt Douglass (North Texas) The free Lie algebra and Schur-Weyl duality.

In this expository talk I will introduce the free Lie algebra and explain several connections with representation theory, especially classical Schur-Weyl duality.

Nov 20: Bill Chin (dePaul) Special biserial coalgebras and quantum SL(2).

We will review some things about the representation theory of comodules, such as Auslander-Reiten theory, and introduce special biserial and string coalgebras. These notions will be applied to comodules over the quantized coordinate Hopf algebra at a root of 1.

Dec 4: Emily Peters (Loyola) The D_2n planar algebras, and knot theories.

I will introduce planar algebras and mostly talk about the example of the D_2n planar algebras. These have the charming feature that most of the calculations you'll want to do with them can be done in a very hands-on, combinatorial way. They also have connections to knot theory, and in particular explain certain coincidences of quantum knot invariants. This is joint work with Scott Morrison and Noah Snyder.